Section 15.9 Stocks and Risky Bonds

Problem. The Value of the Firm (V) is $100 million, the Face Value of the Debt (B) is $80 million, the riskfree rate (r ) is 8.0%, the time to maturity (T) is 1.00 years, and the standard deviation of the return on the firm’s assets () is 30.0%. What is the firm’s Equity Value (E) and Risky Debt Value (D)?

Solution Strategy. Calculate the value of a call and put option on the firm’s asset value, where the face value of the bond’s debt is the option exercise price. There are two methods for calculating the firm’s Equity Value (E) and Risky Debt Value (D). We will use both methods and compare the results.

In the first method, equity is considered to be a call option. Thus, E = Call. The call option is based on the Value of the Firm (V) and the exercise price is the face value of the debt (B). Hence, the call price is calculated from the Black-Scholes call formula by substituting V for S and B for E. The rational is that if V > B, then the equityholders gain the net profit V-B. However, if V < B, then the equityholders avoid the loss by declaring bankruptcy, turning V over to the debtholders, and walking away with zero rather than owing money. Thus, the payoff to equityholders is Max (V - B, 0), which has the same payoff form as a call option. Further, we can use the fact that Risky Debt plus Equity equals Total Value of Firm (D + E = V) and obtain D = V – Call.

In the second method, Risky Debt is considered to be Riskfree Debt minus a Put option. Thus, D = Riskfree Debt - Put. The put option is based on the Value of the Firm (V) and the exercise price is the face value of the debt (B). Hence, the put price is calculated from the Black-Scholes put formula by substituting V for S and B for E. The rational is that the put option is a Guarantee against default in repaying the face value of the debt (B). Specifically, if V > B, then the equityholders repay the face value B in full and the value of the guarantee is zero. However, if V < B, then the equityholders only pay V and default on the rest, so the guarantee must pay the balance B - V. Thus, the payoff on the guarantee is Max (B - V, 0), which has the same payoff form as a put option. Further, we can use the fact that Risky Debt plus Equity equals Total Value of Firm (D + E = V) and obtain E = V - Riskfree Debt + Put.

FIGURE 15.9.1 Spreadsheet for Stocks and Risky Bonds.


How To Build Your Own Spreadsheet Model.

  1. Inputs. Enter the inputs described above into the range B4:B8.
  1. d1 and d2 Formulas. The formula is . In cell B11, enter

=(LN(B4/B5)+(B6+B8^2/2)*B7)/(B8*SQRT(B7))

The formula is . In cell B12, enter

=B11-B8*SQRT(B7)

  1. Cumulative Normal Formulas. Enter using the cumulative normal function in cell B13

=NORMSDIST(B11)

Copy the cell B13 to cell B14 or enter using the cumulative normal function in cell B14

=NORMSDIST(B12)

  1. Black-Scholes Formulas. Translating the notation to this particular application, the Black-Scholes Call formula is . In cell B15, enter

=B13*B4-B14*B5*EXP(-B6*B7)

Similarly, the Black-Scholes Put formula is In cell B16, enter

=(B13-1)*B4+(1-B14)*B5*EXP(-B6*B7)

  1. Riskfree Debt Value. The present value of riskfree debt paying at maturity is . Enter =B5*EXP(-B6*B7) in cell B17.
  1. Method One. Based on the first method:
  • Equity = Call. Enter =B15 in cell C22.
  • Risky Debt = V – Call. Enter =B4-B15 in cell C24.
  • Total Value = Equity + Risky Debt. Enter =C22+C24 in cell C26.
  1. Method Two. Based on the second method:
  • Equity = V – Riskfree Debt + Put. Enter =B4-B17+B16 in cell F22.
  • Risky Debt = Riskfree Debt – Put. Enter =B17-B16 in cell F24.
  • Total Value = Equity + Risky Debt. Enter =F22+F24 in cell F26.

Both methods of doing the calculation find that the Equity Value (E) = $28.24 and the Risky Debt Value (D) = $71.76. We can verify that both methods should always generate the same results. Suppose that we equate the two expressions for the Equity Value (E) to obtain Call = V – Riskfree Bond + Put. Clearly, this is the Put-Call Parity Relationship with the appropriate variable substitutions. Similarly, suppose we equate the two expressions for the Risky Debt Value (D) to obtain V - Call = Riskfree Bond - Put. This is simply a rearrangement of the same Put-Call Parity equation.

Using The Power Of Your Spreadsheet Model.

What would happen to the firm’s Equity Value, Risky Debt Value, and Risky Debt Yield if you increased Firm Asset Standard Deviation? Interestingly, an increase in Standard Deviation causes equityholders to win and the debtholders to lose (see the graph in Figure 15.9.2). Note that this analysis holds the Total Value of the Firm constant, so this is strictly a transfer of wealth from debtholders to equityholders.

FIGURE 15.9.2 Spreadsheet of the Sensitivity of Equity Value, Risky Debt Value, & Risky Debt Yield.


(1)Output Formula for the Risky Debt Yield. The relationship between Risky Debt Value (D) and Risky Debt Yield (Y) is . Solving for the Risky Debt Yield, we get . Enter =-LN(C24/B5)/B7 in cell B33.

(2)Create A List of Input Values and Add Two More Output Formulas. Create a list of input values for the Firm Asset Standard Deviation (10.0%, 20.0%, 30.0%, etc.) in the range C30:G30. Add two more output formulas. One that references the firm’s Equity Value (E) by entering the formula =C22 in cell B31. Another that references the firm’s Risky Debt Value (D) by entering the formula =C24 in cell B32.

(3)Data Table. Select the range B30:G33 for the Data Table. This range includes both the list of input values at the top of the data table and the three output formulas on the side of the data table. Then choose Data Table from the main menu and a Table dialog box pops up. Enter the Firm Asset Standard Deviation cell B8 in the Row Input Cell and click on OK.

(4)Graph the Sensitivity Analysis. Highlight the range C30:G32 and then choose Insert Chart from the main menu. Select an XY(Scatter) chart type and make other selections to complete the Chart Wizard.

It may seem surprising that an increase in standard deviation causes equityholders to win and debtholders to lose, but this is a direct consequence equity being a call option and debt being V minus a call option. We know that increasing the standard deviation makes a call more valuable, so equivalently it increases Equity Value and decreases Debt Value.

The intuitive rational for this is that an increase in standard deviation allows equityholders to benefit from more frequent and bigger increases in V, while not being hurt by more frequent and bigger decreases in V. In the later case, the equityholders are going to declare bankruptcy anyway so they don’t care how much V drops. Debtholders are the mirror image. They do not benefit from more frequent and bigger increases in V since repayment is capped at B, but they are hurt by more frequent and bigger decreases in V. In the latter case, the size of the repayment default (B – V) increases as V drops more. More frequent and bigger defaults cause the Risky Debt Yield (Y) to increase. Looking at the Date Table, we see that as the standard deviation increases from 10% to 50%, the Risky Debt Yield increases from 8.0% (actually, 8.004%) to a huge 18.0%.

The possibility of transferring wealth from debtholders to equityholders sows the seeds of conflict between equityholders and debtholders. Equityholders would like the firm to take on riskier projects, but debtholders would like the firm to focus on safer projects.